Detailed characterization of natural and induced fractures away from the borehole, reliable sequestration of CO2, assessing mechanical integrity and predicting borehole breakout in oil and gas reservoirs, and probing cement integrity require new approaches to material characterization and imaging. Among the existing probes of material elasticity, the elastic nonlinear response (anelasticity) is an approach to probe material integrity. The elastic nonlinearity of a solid is a measure of the material mechanical damage state. Elastic nonlinear response tends to be the best measure of material damage. Thus, as a near-borehole probe, nonlinear methods hold the most promise for discerning mechanically damaged regions or potential crack/fracture conduits.